Grade 12 : Mathematics - Relations (Set 5) | Concept of Maths

Question 1

1. Let $R$ be a relation on the set of real numbers $\mathbb{R}$ defined by $aRb \iff 1 + ab > 0$. Which of the following properties does $R$ satisfy?

Accepted Answer

C) Reflexive and Symmetric

Answer

A) Reflexive and Transitive

Answer

B) Symmetric and Transitive

Answer

D) Equivalence Relation

Question 2

2. Let $A = \{1, 2, 3, 4, 5\}$. The number of equivalence relations on $A$ that contain the pair $(1, 2)$ is:

Accepted Answer

B) 15 (Bell number context)

Answer

A) 15

Answer

C) 52

Answer

D) 20

Question 3

3. Consider the relation $R = \{(a, b) : a, b \in \mathbb{Z}, |a^2 - b^2| < 1\}$. This relation is:

Accepted Answer

D) An Identity Relation

Answer

A) Transitive and Symmetric but not Reflexive

Answer

B) Reflexive and Symmetric but not Transitive

Answer

C) Symmetric only

Question 4

4. Let $S$ be the set of all straight lines in a plane. Let $R$ be a relation on $S$ defined by $L_1 R L_2$ if $L_1$ is perpendicular to $L_2$. Then $R$ is:

Accepted Answer

A) Symmetric but neither reflexive nor transitive

Answer

B) Symmetric and Transitive

Answer

C) Reflexive and Symmetric

Answer

D) None of these

Question 5

5. If $A = \{x, y, z\}$, the number of possible reflexive relations on $A$ is:

Accepted Answer

C) $2^6$

Answer

A) $2^3$

Answer

B) $2^9$

Answer

D) $3^2$

Question 6

6. Let $R$ be a relation on $\mathbb{N}$ defined by $xRy \iff x+2y = 10$. The range of $R$ is:

Accepted Answer

D) {1, 2, 3, 4}

Answer

A) {2, 4, 6, 8}

Answer

B) {1, 2, 3, 4, 5}

Answer

C) {4, 3, 2, 1}

Question 7

7. Let $R$ be a relation on $A = \{1, 2, 3\}$ given by $R = \{(1,1), (2,2), (1,2), (2,1)\}$. This relation is:

Accepted Answer

B) Symmetric and Transitive but not Reflexive

Answer

A) An Equivalence Relation

Answer

C) Reflexive and Symmetric but not Transitive

Answer

D) Reflexive and Transitive but not Symmetric

Question 8

8. Total number of equivalence relations on a set with 3 elements is:

Accepted Answer

A) 5

Answer

B) 3

Answer

C) 2

Answer

D) 8

Question 9

9. If $R$ and $S$ are two equivalence relations on a set $A$, then $R \cup S$ is:

Accepted Answer

D) Not necessarily an equivalence relation

Answer

A) Always an equivalence relation

Answer

B) Never an equivalence relation

Answer

C) An equivalence relation only if $R \subset S$ or $S \subset R$

Question 10

10. A relation $R$ on $\mathbb{R}$ defined by $aRb \iff a \leq b^2$ is:

Accepted Answer

B) Neither Reflexive, nor Symmetric, nor Transitive

Answer

A) Reflexive

Answer

C) Reflexive and Transitive

Answer

D) Symmetric only

Question 11

11. Let $R$ be a relation on $\mathbb{N}$ defined by $xRy \iff x$ divides $y$. Then $R$ is:

Accepted Answer

C) Reflexive, Transitive and Antisymmetric

Answer

A) Equivalence Relation

Answer

B) Symmetric and Transitive

Answer

D) Reflexive and Symmetric

Question 12

12. The relation $R = \{(1,2), (2,3), (3,1)\}$ on set $\{1,2,3\}$ is:

Accepted Answer

A) Neither reflexive, nor symmetric, nor transitive

Answer

B) Transitive only

Answer

C) Symmetric only

Answer

D) Reflexive only

Question 13

13. Let $R$ be a relation on $\mathbb{R}$ defined by $aRb \iff |a-b| \leq 1$. Then $R$ is:

Accepted Answer

B) Reflexive and Symmetric but not Transitive

Answer

A) An equivalence relation

Answer

C) Symmetric and Transitive but not Reflexive

Answer

D) Reflexive and Transitive but not Symmetric

Question 14

14. If $n(A) = m$, the number of symmetric relations on $A$ is:

Accepted Answer

C) $2^{m(m+1)/2}$

Answer

A) $2^{m^2}$

Answer

B) $2^{m(m-1)/2}$

Answer

D) $m^2 - m$

Question 15

15. Let $R = \{(x, y) : x, y \in \mathbb{R}, x^2 + y^2 = 1\}$. This relation is:

Accepted Answer

D) Symmetric

Answer

A) Reflexive

Answer

B) Transitive

Answer

C) Anti-symmetric

Question 16

16. If $R$ is an equivalence relation on $A$, then $R^{-1}$ is:

Accepted Answer

A) Always an equivalence relation

Answer

B) Only reflexive and symmetric

Answer

C) Never an equivalence relation

Answer

D) Only the identity relation

Question 17

17. Let $L$ be the set of all lines in XY plane and $R$ be the relation in $L$ defined as $L_1 R L_2 \iff L_1$ is parallel to $L_2$. Then $R$ is:

Accepted Answer

C) An Equivalence Relation

Answer

A) Symmetric and Transitive but not Reflexive

Answer

B) Reflexive and Symmetric but not Transitive

Answer

D) None of these

Question 18

18. On the set $\mathbb{Q} \setminus \{0\}$, let $aRb \iff a = 1/b$. This relation is:

Accepted Answer

B) Symmetric

Answer

A) Reflexive

Answer

C) Transitive

Answer

D) Equivalence Relation

Question 19

19. The number of reflexive relations on a set with $n$ elements is $2^{n^2-n}$. For $n=2$, how many such relations exist?

Accepted Answer

A) 4

Answer

B) 16

Answer

C) 8

Answer

D) 2

Question 20

20. Let $A = \{1, 2, 3\}$. Which of the following is the smallest equivalence relation on $A$?

Accepted Answer

C) $\{(1,1), (2,2), (3,3)\}$

Answer

A) $A 	imes A$

Answer

B) $\{(1,1), (2,2)\}$

Answer

D) $\emptyset$